Multiresolution models

One is always trying to develop more and more detailed models for interstellar clouds. In many statistical studies it would be important to have as wide a range of different size scales as possible. One can increase discretization (divide model into smaller and smaller cells) but with three-dimensional models one very rapidly runs against limitations set by available computers. Models require gigabytes of memory and run times become uncomfortably long. There are several ways these problems can be alleviated (see, for example, the section on dust emission calculations). One way becomes apparent when one realizes that a detailed model does not necessarily have to be detailed everywhere.

Interstellar clouds (and models thereof) are extremely inhomogeneous and we are mostly interested in their densest parts. For example, if we study star-formation we want to be able to model the development of the densest cores and the rest of the cloud is less important. Similarly, most molecular line emission comes from dense gas and it could be modelled accurately even if calculations for low-density gas were less precise. This can be achieved with multiresolution models. Cloud is divided into smaller cells only in its densest parts and more crude division is used elsewhere. In addition to density the division can take into account other factors like the velocity field or temperature variations.

We have implemented a program for computing line emission from three-dimensional cloud models that are built from cells of different sizes. The figure below shows one such cloud. Colours indicate gas density and one can see that better discretization (smaller cells) is used in regions of high density. The amount of memory and time needed for radiative transfer calculations is directly proportional to the number of cells. Therefore, multiresolution calculations can give significant savings in run times and still provide full resolution and good accuracy in the most important parts of the models.

a multiresolution model
Cell division in one multiresolution model. The colours indicate the density (yellow=highest density). An animation where the model is rotated and different cross sections are shown is available as a mjpeg-file (6.8MB!!, 320x200) and as a divx-file (2.6MB, 640x480).

The model above contained cells of three different sizes. We have used similar models when computing 13CO spectra from MHD models. Very good accuracy was obtained even when the number of cells was ~80% less than what the model would contain if the best resolution was used everywhere. Therefore, calculations required five times less time and five times less memory. This new program for line transfer is no longer based on Monte Carlo simulation since that becomes rather inefficient when models contain cells of different sizes.

The following picture shows what can be achieved with multi-resolution simulations of magnetohydrodynamic turbulence. This MHD simulation represents an interstellar cloud with a diameter of 5pc. Using a hierarchy of grids with smaller and smaller cell size one can zoom into small, interesting regions. In this case one has followed the formation of some protostellar systems which are resolved with a resolution of a few astronomical units. The simulation was done at UCSD by Padoan et al.

a AMR simulation of a star-forming cloud
A model of an interstellar cloud created using AMR (Automatic Mesh Refinement) techniques. The cloud is discretized using a hierarchy of grids with different grid step sizes. This allows simulation of an entire cloud (several parsecs) and at the same time zooming-in on the development of protostellar systems (resolution of a few AU). The difference in linear scales is about one million. (Padoan, Kritsuk, Norman / UCSD)

The next challenge is to do radiative transfer calculations on such deeply nested grids. Our idea is to carry out the radiative transfer calculations independently on each subgrid of the grid hierarchy. This requires that the intensity entering or exiting a grid is stored at the grid boundary or, in a parallel computer implementation, the grids exchange information about the radiation field intensity at the dividing surface.

The next figure shows an example of the first step in any radiative transfer modelling. Simulation has been done for one frequency, resulting in information about the radiation field strength at each position of the cloud.

infrared dust emission from an interstellar cloud
Radiation intensity at selected wavelength inside a AMR model cloud. The white boxes show the outlines of the main grid and the various subgrids used in the spatial discretization. The density distribution is taken from a MHD simulation made by A. Razoumov (ORNL). In the divx movie (2.5MB) the model is rotated and one flies through cloud.

Once intensity is known at each position of the cloud and for many frequencies, that information can be used to solve dust temperatures and molecular excitation. We have working prototypes of radiative transfer programs for both continuum and line transfer. The next figure shows results from a test run where we have computed infrared dust emission for an interstellar cloud with a diameter of 6 parsecs. Actually, the density distribution is from a cosmological simulation made by A. Razoumov, ORNL. Here that model is scaled to much smaller size and serves only as an illustration of the kind of modelling that is possible also in the case of an interstellar cloud.

infrared dust emission from an interstellar cloud
Infrared dust emission calculated for a three-dimensional cloud that is defined using a three-dimensional, hierarchical AMR grid. The upper right hand frame shows the dust optical depths and the lower right hand frame the corresponding surface brightness of the thermal dust emission. The left hand frame zooms into the densest and most bright region. The black lines indicate the outlines of the different subgrids that are used in the spatial discretization of the density distribution.

The final figure shows an example of line transfer calculations performed on an AMR grid. Again, this is not yet a realistic model of any astronomical object but just illustrates the possibilities presented by the use of hierarchical grids. In this case we have taken one MHD simulation and sampled that onto a regular grid of 403 cells. The AMR grid was created by putting 10 such grids inside each other. For each subgrid the cell size was reduced to half so that the cells of the innermost subgrid are 512 times smaller than the cells of the root grid and the ratio between the sizes of the entire cloud and the smallest cells is about 20000. We performed non-LTE radiative transfer calculations on the resulting AMR grid and computed the emerging line emission for 13CO molecule. The figure shows the integrated line areas. The gas density was increased for each subgrid so that the emission peaks in the centre of the model.

line emission from an interstellar cloud
13CO line area calculated for a molecular cloud that is sampled using a hierarchical AMR grid. This is not a physically realistic model but the figure and the animation serve only to illustrate how AMR grids make it possible to simulate and model very different size scales. The divx animation (2.2MB) starts from outside the cloud and zooms into the innermost subgrid.
In radiative transfer modelling the memory and run times are, in principle, directly proportional to the number of individual cells (grid points) included in the model. These days one can routinely handle models defined on regular grids with up to 2563 grid points. This is true at least at low temperatures where models have to include only a small number of radiative transitions. In the previous example we built an AMR model by stacking ten grids with 403 grid points each. In fact, one can extend the previous model to include 37 levels of hierarchy before the number of cells becomes equal to even 1283. This means that one could easily make a model that covers the entire Galaxy while the smallest cells are well below one astronomical unit ! Of course, the best resolution would only be reached in a correspondingly very small sub-volume of the entire model. In normal cases the range of scales would probably be less extreme and the best resolution would be used in a much larger fraction of the model volume.